Fig 1.
Representative network of the automobile industry in Japan.
Major firms are selected under the following conditions: i) they are connected to Toyota Motor within three degrees of separation, ii) they belong to either the manufacturing or wholesale sectors, iii) they are listed in the first section of the Tokyo Stock Exchange, and iv) They are in the top 40 in terms of sales. The firms thus selected are displayed as nodes and the transactions between them are displayed as arrows. All of the displayed nodes belong to the GSCC component. The size of the nodes is scaled to the sales of the corresponding firm. The color of the nodes distinguishes their industry type; blue and green designate manufacturing and wholesale, respectively.
Table 1.
Walnut structure: The sizes of the different components.
Fig 2.
Visualization of the network in three-dimensional space.
A surface view of the network is shown in panel (a), and a cross-sectional view that is cut through its center is shown in panel (b). The red, green, and blue dots represent firms in the IN, GSCC, and OUT components, respectively.
Fig 3.
The production network as a walnut structure. The area of each component is approximately proportional to its size.
Table 2.
Walnut structure: The shortest distance from GSCC to IN/OUT.
Table 3.
Modular level statistics.
Fig 4.
Hierarchical structure of the communities.
Five levels of hierarchical community decomposition are illustrated. The width of the triangle originating in each community at the n-th level is proportional to the number of its subcomunities at the (n + 1)-th level.
Fig 5.
Hierarchical decomposition of the whole network into communities and subcommunities.
This panel (a) highlights the 6 largest communities at the top modular level with different colors. Each of these communities is further decomposed into subcommunities as demonstrated in panels (b) through (g), where the 6th largest subcommunities of the 1st through the 6th largest communities are highlighted.
Fig 6.
The complementary cumulative distribution function D(s) of the community size s at the top modular level.
Fig 7.
Polarizability of the direction of links interconnecting communities at the top level.
Here, 51 major communities containing more than 1,000 firms are selected. The top figure plots the polarization ratio |Pij| of the linkage between communities i and j versus the total number Lij of its constituting links. The dashed curve shows the significance level corresponding to 2σ for the polarizability of intercommunity linkage for the given total number of its constituents, where the random orientation of the individual links is adopted as a null model; see Eq (6) for the standard deviation σ. The bottom figure is a histogram for the frequency of intercommunity linkages in each bin of Lij. The grey (black) bars depict the number of intercommunity linkages with a |Pij| that is higher (lower) than the threshold for the test of statistical significance.
Table 4.
Overexpressions of the 1st level communities.
Fig 8.
Network of the 50 largest communities at the top level.
The major communities are depicted as nodes, and their size is scaled to the size of their corresponding communities. A bundle of directed links connecting a pair of nodes in either direction is represented by an arrow, the width of which is proportional to the total number of their links.
Fig 9.
(color online) The complementary cumulative distribution function D(s) of a community with size s at the second modular level.
A power-law fit to the data (red line) using the maximum likelihood estimation technique yields D(s) ∼ s−γ+1 with γ = 2.50 ± 0.02, smin = 28.2 ± 7.6, and p value = 0.976.
Fig 10.
Overexpression network of sectors.
The node size represents the percentage of firms belong to that particular sector.
Fig 11.
The complementary cumulative distribution of link-weight in the overexpression network.
Table 5.
Top five heaviest weighted links between sectors.
Fig 12.
Polarizability of the direction of the links interconnecting communities at the second level.
Here, 1086 communities containing over 100 firms are selected. The dashed curve represents the same significance level as in Fig 7.
Fig 13.
Triangular diagram classifying communities at the second level by their relationship with the walnut structure.
Each community is depicted by a circle located at point (x, y) inside the equilateral triangle, which corresponds to the composition (x1, x2, and x3) of firms belonging to the IN, OUT, and GSCC components that are represented in three-dimensional space; the one-to-one correspondence between (x, y) and (x1, x2, x3) is illustrated in the associated figure (a). The size of the communities is reflected by the area of their associated circles. The triangular region is decomposed into six domains with the average composition (,
,
) of the IN, OUT, and GSCC components for all firms, as designated in the associated figure (b); see the text for more detailed information on the domain decomposition.
Table 6.
Classification of communities at the second level based on the walnut structure.
Fig 14.
Density of links over intergroups.
These figures show how many links the intergroups have. The top figure (a) shows the 3D plots of the industrial sectors. The bottom figure (b) shows the 3D plots of the communities.